John Tate
Encyclopedia
John Torrence Tate Jr. (born March 13, 1925) is an American
mathematician
, distinguished for many fundamental contributions in algebraic number theory
, arithmetic geometry and related areas in algebraic geometry
.
, and a longtime editor of Physical Review
. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University
, and entered the doctoral program in physics at Princeton University
. He later transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin
. Tate taught at Harvard for 36 years before joining the University of Texas in 1990. He retired from the Texas mathematics department in 2009, and currently resides in Cambridge, Massachusetts
with his wife Carol. He has three daughters with his first wife Karin Tate.
(1950) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring
, its self-duality and harmonic analysis on it; independently and a little earlier, Kenkichi Iwasawa
obtained a similar theory. Together with his teacher Emil Artin
, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology
applied to the idele class group and Galois cohomology
. This treatment made more transparent some of algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group
of a global field.
Subsequently Tate introduced what are now known as Tate cohomology group
s. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory
. With Jonathan Lubin
, he recast local class field theory by the use of formal group
s, creating the Lubin–Tate local theory of complex multiplication
.
He has also made a number of individual and important contributions to p-adic theory; for example, Tate's invention of rigid analytic space
s can be said to have spawned the entire field of rigid analytic geometry. He found a p-adic analogue of Hodge theory
, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory
. Other innovations of his include the 'Tate curve' parametrization for certain p-adic elliptic curve
s and the p-divisible (Tate–Barsotti) groups
.
Many of his results were not immediately published and some of them were written up by Serge Lang
, Jean-Pierre Serre
, Joseph H. Silverman
and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties. The classification of abelian varieties over finite field
s was carried out by Taira Honda and Tate (the Honda–Tate theorem).
The Tate conjecture
s are the equivalent for étale cohomology
of the Hodge conjecture
. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings
.
Tate has also had a major influence on the development of number theory through his role as a
Ph.D. advisor. His students include Joe Buhler, Benedict Gross
, Robert Kottwitz, James Milne, V. Kumar Murty, Carl Pomerance
, Ken Ribet, Ted Chinburg, Joseph H. Silverman
, Dinesh Thakur, Jeremy Teitelbaum.
for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society. He was awarded a Wolf Prize in Mathematics
in 2002/03 for his creation of fundamental concepts in algebraic number theory
.
In 2010, the Norwegian Academy of Science and Letters
, of which he is a member, awarded him the Abel Prize
, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory
and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics."
Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner
, Chairman of the Department of Mathematics at the University of Texas.
Ph.D. thesis under Emil Artin
. Reprinted in
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, distinguished for many fundamental contributions in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, arithmetic geometry and related areas in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
Biography
Tate was born in Minneapolis. His father, John Tate Sr., was a professor of physics at the University of MinnesotaUniversity of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
, and a longtime editor of Physical Review
Physical Review
Physical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, and entered the doctoral program in physics at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. He later transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
. Tate taught at Harvard for 36 years before joining the University of Texas in 1990. He retired from the Texas mathematics department in 2009, and currently resides in Cambridge, Massachusetts
Cambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...
with his wife Carol. He has three daughters with his first wife Karin Tate.
Mathematical work
Tate's thesisTate's thesis
In number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....
(1950) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
, its self-duality and harmonic analysis on it; independently and a little earlier, Kenkichi Iwasawa
Kenkichi Iwasawa
Kenkichi Iwasawa was a Japanese mathematician who is known for his influence on algebraic number theory.Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture...
obtained a similar theory. Together with his teacher Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
applied to the idele class group and Galois cohomology
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
. This treatment made more transparent some of algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
of a global field.
Subsequently Tate introduced what are now known as Tate cohomology group
Tate cohomology group
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence...
s. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
. With Jonathan Lubin
Jonathan Lubin
Jonathan Darby Lubin is a mathematician who introduced Lubin–Tate formal group laws.-References:**...
, he recast local class field theory by the use of formal group
Formal group
In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between...
s, creating the Lubin–Tate local theory of complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
.
He has also made a number of individual and important contributions to p-adic theory; for example, Tate's invention of rigid analytic space
Rigid analytic space
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. They were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group...
s can be said to have spawned the entire field of rigid analytic geometry. He found a p-adic analogue of Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
. Other innovations of his include the 'Tate curve' parametrization for certain p-adic elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s and the p-divisible (Tate–Barsotti) groups
Barsotti–Tate group
In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p...
.
Many of his results were not immediately published and some of them were written up by Serge Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
, Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, Joseph H. Silverman
Joseph H. Silverman
Joseph Hillel Silverman is currently a professor of mathematics at Brown University. Joseph Silverman received an Sc.B. from Brown University in 1977 and a Ph.D. from Harvard University in 1982 under the direction of John Tate. He taught at M.I.T...
and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties. The classification of abelian varieties over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s was carried out by Taira Honda and Tate (the Honda–Tate theorem).
The Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
s are the equivalent for étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
of the Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings
Gerd Faltings
Gerd Faltings is a German mathematician known for his work in arithmetic algebraic geometry.From 1972 to 1978, he studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathematics and in 1981 he got the venia legendi in mathematics, both from the University...
.
Tate has also had a major influence on the development of number theory through his role as a
Ph.D. advisor. His students include Joe Buhler, Benedict Gross
Benedict Gross
Benedict Hyman Gross is an American mathematician, the George Vasmer Leverett Professor of Mathematics at Harvard University and former Dean of Harvard College....
, Robert Kottwitz, James Milne, V. Kumar Murty, Carl Pomerance
Carl Pomerance
Carl Bernard Pomerance is a well-known number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least 7 distinct prime factors. He immediately joined the faculty at the...
, Ken Ribet, Ted Chinburg, Joseph H. Silverman
Joseph H. Silverman
Joseph Hillel Silverman is currently a professor of mathematics at Brown University. Joseph Silverman received an Sc.B. from Brown University in 1977 and a Ph.D. from Harvard University in 1982 under the direction of John Tate. He taught at M.I.T...
, Dinesh Thakur, Jeremy Teitelbaum.
Awards and honors
In 1956 Tate was awarded the American Mathematical Society's Cole PrizeCole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...
for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society. He was awarded a Wolf Prize in Mathematics
Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
in 2002/03 for his creation of fundamental concepts in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
.
In 2010, the Norwegian Academy of Science and Letters
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
, of which he is a member, awarded him the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics."
Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner
William E. Beckner
-External links:*...
, Chairman of the Department of Mathematics at the University of Texas.
Selected publications
, Princeton UniversityPrinceton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
Ph.D. thesis under Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
. Reprinted in
- Tate, John (1952), " The higher dimensional cohomology groups of class field theory ", Annals of Mathematics 56 : 294-297, MR 0049950.
- Tate, John (1963), " Algebraic cycles and poles of zeta functions", in Arithmetical Algebraic Geometry, Harper and Row: 93-110, MR0225778.
- Tate, John (1984), " Les conjectures de Stark sur les fonctions L d'Artin en s=0 ", Progress in Math. 47, Birkhäuser Boston, MR0782485.
See also
- Barsotti–Tate groupBarsotti–Tate groupIn algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p...
- Néron–Tate heightNéron–Tate heightIn number theory, the Néron–Tate height is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field...
- Sato–Tate conjecture
- Tate dualityTate dualityIn mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and .-Local Tate duality:...
- Tate twistTate twistIn number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules.For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutK is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers,...
- Tate's algorithm